MIAMI — Like many Cuban baseball stars, outfielder Leonys Martin dreamed of leaving the communist island for the bright lights and big money of Major League Baseball. Martin accomplished his goal in 2011 when he signed with the Texas Rangers, but not before what court documents and the Justice Department describe as a harrowing ordeal in which he was held for ransom in Mexico while his family members were kept under surveillance in South Florida.

Three people have been indicted in Miami on federal charges of hostage-taking and extortion conspiracy — counts that carry potential life prison sentences if they are convicted — and Martin himself is suing his alleged kidnappers for the return of more than $1.3 million he has already paid them.

Andrew Zimbalist, a Smith College professor who specializes in sports economics, said the Martin case appears unique.

“I have never heard of a kidnapping case like this,” Zimbalist said.

Many Cubans have defected over the years to play ball in the U.S., including such current stars as Cincinnati Reds pitcher Aroldis Chapman, Los Angeles Dodgers outfielder Yasiel Puig and Oakland Athletics outfielder Yoenis Cespedes. In September, the Cuban government reversed decades of policy by announcing that its athletes will be allowed to sign contracts to compete in foreign leagues without defecting.

Compared the vast majority of immigrants, Cubans get unique treatment by the U.S. government under the “wet foot, dry foot” policy that came about during the Cold War. Cubans who reach U.S. soil are generally allowed to stay in this country, while those intercepted outside the U.S. are sent back home.

Martin, 25, a speedy center fielder for the Rangers, made his MLB debut in September 2011. Last season, in 147 games, he hit .260 with 49 runs batted in and 36 stolen bases. Martin and his attorney, Paul Minoff, declined comment for this story, as did the Rangers and MLB officials, citing the ongoing litigation.

A few years before joining Texas, Martin was a rising star in Cuba, traveling to baseball tournaments around the globe with the national team. After an August 2010 tournament in Japan, according to his lawsuit, he decided to leave for the U.S.

He and several family members and friends made contact with a man who offered them a trip from Cuba on a yacht to Cancun, Mexico. From there, they could eventually cross by land into the U.S. But instead of journeying directly north, they were taken to a house lined inside with mattresses and watched by two armed men, one of them identified as Eliezer Lazo.